|
%I #29 Feb 26 2018 09:18:01
%S 390503,480803,351121,566821,78901,578701,323623,606223,326701,645901,
%T 619471,745471,655717,842617,437321,855821,854713,876913,811337,
%U 915437,561409,920509,515401,956401,452401,1023601,805633,1049333,247141,1092541,1037903,1127603
%N First pair of primes (p1, p2) that begin centuries of primes having the same prime configuration, ordered by increasing p2. Each configuration is allowed only once.
%C Rearranged the pairs of numbers so that the sequence of values of p2 increases. The first pair is for the primes 390500 + {3, 27, 39, 53, 81} and 480803 + {3, 27, 39, 53, 81}. There is a large, but finite number of terms. How many terms are there? - _T. D. Noe_, Feb 10 2013
%C The sequence lists the small prime twin centuries. As exploration goes into higher primes many are found to be triples or even higher. Example: 1072009 is a twin with 5179509, a triple with 7183109, quadruple with 8284709, quintuple with 8462609, and sextuple with 9739309, and there could be infinitely more. - _Ki Punches_, Dec 17 2009
%C The first two centuries without any primes start with 1671800 and 2637800. These are not included in the sequence since they do not have a first prime. However, if they were to be included they would be the 136th pair. - _Andrew Howroyd_, Feb 25 2018
%H T. D. Noe, <a href="/A164987/b164987.txt">Table of n, a(n) for n = 1..10000</a>
%e The primes in 480800..480899 are 480803, 480827, 480839, 480853, 480881 ending with 03, 27, 39, 53, 81. The primes in 390500..390599 end with the same digits, and no earlier pair has this quality. Hence a(1) = 390503 and a(2) = 480803.
%t pSig[n_] := Prime[Range[PrimePi[100 n] + 1, PrimePi[100 (n + 1)]]] - 100 n; t = {}; c = {}; found = {}; Do[s = pSig[n]; If[Length[s] > 0 && ! MemberQ[found, s] && MemberQ[c, s], d = Mod[s[[1]], 100]; AppendTo[found, s]; AppendTo[t, {Position[c, s][[1, 1]]*100 + d, n*100 + d}]]; AppendTo[c, s], {n, 11000}]; Flatten[t] (* _T. D. Noe_, Feb 10 2013 *)
%o (PARI)
%o sig(c)={my(s=0); for(v=0,49,if(isprime(100*c+2*v+1),s+=2^v)); s}
%o pairs(n)={my(L=List(),M=Map(),c=0); while(#L<2*n, c++; my(s=sig(c),f=0); if(mapisdefined(M,s,&f), if(f&&s,my(d=2*valuation(s,2)+1); listput(L,100*f+d); listput(L,100*c+d); mapput(M,s,0)), mapput(M,s,c))); Vec(L)}
%o pairs(20) \\ _Andrew Howroyd_, Feb 25 2018
%Y Cf. A038822 (number of primes between 100n and 100n+99).
%Y Cf. A181098, A186393-A186408 (centuries having 0 to 16 primes).
%K base,nonn,fini
%O 1,1
%A _Ki Punches_, Sep 03 2009 through Dec 06 2009
%E Terms rearranged by _T. D. Noe_, Feb 10 2013
|
